= Odd-cycle copies from positive triangle density
{title2=$\#K_3\ge\beta n^3\Longrightarrow\#C_\ell\ge\zeta n^\ell$}
For any fixed odd length $\ell\ge3$, positive <triangle in a graph> density forces positive density of $\ell$-cycles. A <Szemerédi regularity lemma> partition retains a <triangle in a graph> of regular dense pairs after the sparse and exceptional pairs are removed. Count embeddings of a proper three-colouring of $C_\ell$ into those three clusters. The constants depend on $\beta$ and $\ell$, not on <graph> order.
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